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Arcsine laws and interval partitions derived from a stable subordinator. (English) Zbl 0769.60014
Let \((T_ s: s\geq 0)\) be a subordinator, i.e. an increasing process with independent increments, stable with index \(\alpha\), \(0<\alpha<1\), and let \((B_ t: t\geq 0)\) be a process whose zero set is the closed range of the subordinator. Define \(\Gamma_ +(u)\) to be the time \((B_ t)\) is positive for \(0\leq t<u\). It is shown that, for each \(u>0\), the distributions of \(\Gamma_ +(u)/u\) and \(\Gamma_ +(T_ s)/T_ s\) are the same. This is a generalization of Lévy’s result in which \((B_ t)\) is a Brownian motion and \((T_ s)\) is the inverse of the continuous local time process associated with the set of zeros of \((B_ t)\). This identity in distribution is extended to a large class of functionals derived from the lengths and signs of excursions of \((B_ t)\) away from 0 for the classical Brownian motion setup, and similar identities are obtained in the general case of a process whose zero set is the range of a stable subordinator.

MSC:
60E07 Infinitely divisible distributions; stable distributions
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J99 Markov processes
60J55 Local time and additive functionals
60J65 Brownian motion
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